| Abstract |
We prove the geometric Breuil--M\'ezard conjecture for two-dimensional semi-stable representations of $G_{\mathbf{Q}_{p^2}}$ with parallel Hodge--Tate weights $(0,r)$, where $r$ is odd and lies in the Fontaine--Laffaille range. Our strategy is to prove the numerical Breuil--M\'ezard conjecture at a maximally non-split mod-$p$ representation on each irreducible component of the special fiber of the Emerton--Gee stack. Building on previous work on strongly divisible modules, we obtain explicit descriptions of the relevant semi-stable deformation rings at these points and use them to establish the numerical Breuil--M\'ezard conjecture. We then explain how these local results lead to the geometric Breuil--M\'ezard conjecture in this setting. This is joint work with Seongjae Han and Hyeonmin Kim. |
